Question

A function is said to vanish to order at if the limit lim→()(−) exists (and is not infinite). The order of vanishing of a function quantifies the rate at which ()→0 as →; for example, if vanishes to order 3 at 2 then () approaches zero at least as quickly as does (−2)3 as →2.

Answer 1

The answer is "vanishes to order 3 at a=1"

Explanation:

Please find the complete question in the attached file.

`\to f(x) =(1)/(x) -x^2-3x+3\\\\\to f(x)= (1-x^3+3x^2-3x)/(x)\\\\\to f(x)= ((1-x)^3)/(x) = - ((x-1)^3)/(x)\\\\\to \lim_(x \to 1) f(x) = \lim_(x \to 1) ((1-x)^3)/(x)`

vanishes to order 3 at a=1.